Year: 2013
Paper: 2
Question Number: 1
Course: LFM Pure
Section: Differentiation
All questions were attempted by a significant number of candidates, with questions 1 to 3 and 7 the most popular. The Pure questions were more popular than both the Mechanics and the Probability and Statistics questions, with only question 8 receiving a particularly low number of attempts within the Pure questions and only question 11 receiving a particularly high number of attempts.
Difficulty Rating: 1600.0
Difficulty Comparisons: 0
Banger Rating: 1484.0
Banger Comparisons: 1
\begin{questionparts}
\item Find the value of $m$ for which the line
$y = mx$ touches the curve $y = \ln x\,$.
If instead the line intersects the curve when $x = a$ and $x = b$, where $a < b$, show that $a^b = b^a$.
Show by means of a sketch that $a < \e < b$.
\item The line $y=mx+c$, where $c>0$, intersects the curve $y=\ln x$ when $x=p$ and $x=q$, where $p < q$. Show by means of a sketch, or otherwise, that $p^q > q^p$.
\item Show by means of a sketch that the straight line through the points $(p, \ln p)$ and $(q, \ln q)$, where $\e\le p < q\,$, intersects the $y$-axis at a positive value of $y$. Which is greater, $\pi^\e$ or $\e^\pi$?
\item Show, using a sketch or otherwise, that if $0 < p < q$ and $\dfrac{\ln q - \ln p}{q-p} = \e^{-1}$, then $q^p > p^q$.
\end{questionparts}\begin{questionparts}
\item The tangent to $y = \ln x$ is
\begin{align*}
&& \frac{y - \ln x_1}{x - x_1} &= \frac{1}{x_1} \\
\Rightarrow && \frac{x_1y -x_1 \ln x_1}{ x- x_1} &= 1 \\
\Rightarrow && x_1 y - x_1 \ln x_1 &= x - x_1
\end{align*}
So to run through the origin, we need $\ln x_1 = 1 \Rightarrow x_1 = e$ so the line will be $y = \frac1{e} x$
If $ma = \ln a \Rightarrow m = \frac{\ln a}{a} = \frac{\ln b}{b} \Rightarrow b \ln a = a \ln b \Rightarrow a^b = b^a$.
\item
This was the most popular of all of the questions. Overall part (i) of this question was well answered, although there were a number of candidates who were not able to find the tangent and intercept even in this first case. Very few attempts at part (ii) of this question involved the use of sketches. While many attempts at part (iii) recognised the link in the final part with part (ii) of the question, many of the explanations in this section were not well enough explained to gain full marks. In the final part it was pleasing to note that many candidates realised that the conditions implied that the intersection with the y-axis was at a negative value.